# Logic Seminar - Paul Shafer (Leeds)

Dates: | 25 May 2022 |
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Times: | 15:00 - 16:00 |

What is it: | Seminar |

Organiser: | Department of Mathematics |

Who is it for: | University staff, External researchers, Adults, Alumni, Current University students |

Title: Cohesive powers of linear orders

Abstract: A cohesive power of a computable structure is an effective analog of an ultrapower, where a cohesive set acts as an ultrafilter. We compare the properties of cohesive powers to those of classical ultrapowers. In particular, we investigate what structures arise as the cohesive power of B over C, where B varies over the computable copies of some fixed computably presentable structure A and C varies over the cohesive sets.

Let omega, zeta, and eta denote the respective order-types of (N, <), (Z, <), and (Q, <). We take omega as our computably presentable structure, and we consider the cohesive powers of its computable copies. If L is a computable copy of omega that is computably isomorphic to the standard presentation (N, <), then all of L's cohesive powers have order-type omega + (zeta x eta), which is familiar as the order-type of countable non-standard models of PA.

We show that it is possible to realize a variety of order-types other than omega + (zeta x eta) as cohesive powers of computable copies of omega. For example, we show that there is a computable copy L of omega whose power by any Delta_2 cohesive set has order-type omega + eta. More generally, we show that it is possible to achieve order-types of the form omega + certain shuffle sums as cohesive powers of computable linear orders of type omega.